IE 360 Statistical Forecasting and Time Series

Homework 2

Fatih Mehmet Yılmaz - 2017402066

11.05.2022

Part 1 The plot of the time series
Part 2 Autocorrelation functions plot of the time series
Part 3 Adding New Variables
Part 4
As it can be seen, I tried 4 different models. It would be wise to use trend and seasonality variables together. If I were to use X(t-4) variable with these trend and quarter variables, there would be an improvement in the sense that residual standard error decreases and R-squared values increased. However, because 8 observations deleted rather than 4 observations deleted in the case of only trend and quarter variables, there is a trade-off between the loss of data and the improvement in the statistics. Because the dataset is relatively small, this loss of the data is something I would like not to do.
Therefore, I can use the model with the trend and quarter variables without the lagged Xt_4 variable. (Just for the beginning in this step)
Even though using all variables improved the statistics, there are many insignificant variables in the output. Therefore, I will eliminate the variables which seems unrelated in terms of my way of thinking and the statistical terms.
With this result, I prefer not to use GNP_a, GNP_c and GNP_t variables. There is still insignificance in the model.
With these outputs, I can remove these 3 variables that do not contribute to the model meaningfully.
From this point, I will remove Xt_4 variable as well because above, I already decided not to use this variable in my final model.
As compared to the model with trend and quarter seasonality variables, this model works better in terms of residuals and significance of coefficients. Therefore, this is my final model as of now.
It seems that when I use PU and NUGV together, one of the variables becomes insignificant. Therefore, I will not use NUGV which seemed meaningful at the very beginning, I will continue my model with the trend, quarter and independent PU variable.
Above, as you can see, I tried several models. Firstly, I tried trend, quarter (seasonal) and Xt_4 (lagged) variables and I decided to continue with trend and quarter variables. Then, I checked the model with all variables. I started to eliminate unrelated independent variables from the model one by one by checking the residuals and significance. I decided to continue with a model that contains trend, seasonality and independent PU variables.

Part 5 Showing the final model and my explanation

My final model is shown above. As it can be seen, Residuals follows the assumptions. (0-mean, normality and not autocorrelated) Also, the autocorrelation seems very nice. There is no pattern in the residuals plot. If we look at the summary output, I can say that all variables I used (trend, PU and seasonality) are statistically significant. The first quarter is dropped due to the intercept. Also, the p-value obtained by the Breusch-Godfrey test for serial correlation is not small enough. Thus, this model seems reliable.

Part 6 Forecast

Plot of the Predicted Values with the first 6 years:

As seen from the plots, the prediction seems reasonable. Althought the model can be improved by adding some new variables, I think that I reached a decent model within the homework and given variables.